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Not Pascal's Triangle, but does it have name or inventor? "Apex (playing-)Card Trick"

ink1

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Feb 5, 2018
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4
Not Pascal's Triangle, but does it have name or inventor? "Apex (playing-)Card Trick"

Hi

I thought this was a Pascal triangle - but I don't believe it is now. Or at least I know it is NOT 'the' Pascal triangle... maybe it was another variant he came up with.

Does anyone know who or where this came from?

The triangle is made up in effect the opposite way to Pascal's - in that the number above is the sum of the two below

ALSO 9's are cast out [so if the numbers below were 7 + 7... the number above those two would be 5 and not a double digit number - i.e. 14]

Example:
9
8 1
7 1 9
4 3 7 2
3 1 2 5 6

Anyone have any origin info - or background to anything I can bring up in relation to it's history [?]

Thanks for any help
 

Dr.Peterson

Elite Member
Joined
Nov 12, 2017
Messages
2,743
I thought this was a Pascal triangle - but I don't believe it is now. Or at least I know it is NOT 'the' Pascal triangle... maybe it was another variant he came up with.

Does anyone know who or where this came from?

The triangle is made up in effect the opposite way to Pascal's - in that the number above is the sum of the two below

ALSO 9's are cast out [so if the numbers below were 7 + 7... the number above those two would be 5 and not a double digit number - i.e. 14]

Example:
9
8 1
7 1 9
4 3 7 2
3 1 2 5 6

Anyone have any origin info - or background to anything I can bring up in relation to it's history [?]

Thanks for any help
Can you tell us more about this?

Why do you think it would have a name or a history? What significance does it have?

Where did YOU see it, and what was the context? How was it created?

To me, it looks like you either started with a random row of digits on the bottom and added with casting out nines (or, almost, added mod 9) to get the rows above, or else started with the 9 at the top and randomly chose addends in the row below. Either way, it seems to be full of randomness, and therefore not particularly interesting in itself.
 

ink1

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Joined
Feb 5, 2018
Messages
4
Can you tell us more about this?

Why do you think it would have a name or a history? What significance does it have?
You may be correct that there is no history

You are correct in that it may not be interesting

You are 100% correct in that it starts with a random set of digits - I would start with Ten digits along the base

But there are some properties of the ten digit random base that makes this interesting [for me] - I am using this as a 'performance piece' - so was hoping to use some speil at the beginning about it's origins [along with explaining how it works] - before someone selects the 10 random numbers for the base

I know this traingle has been around since the 1950's [at least - as at that point the use of it was not claimed to be new or original]

If there is obviously no history - then I may have to fudge it a bit and speak about Pascal's work on his triangle - and never say this was his [but just glide into it] argh... that may be the best way forward

Thanks for your help
 

Subhotosh Khan

Super Moderator
Staff member
Joined
Jun 18, 2007
Messages
18,086
You may be correct that there is no history

You are correct in that it may not be interesting

You are 100% correct in that it starts with a random set of digits - I would start with Ten digits along the base

But there are some properties of the ten digit random base that makes this interesting [for me] - I am using this as a 'performance piece' - so was hoping to use some speil at the beginning about it's origins [along with explaining how it works] - before someone selects the 10 random numbers for the base

I know this traingle has been around since the 1950's [at least - as at that point the use of it was not claimed to be new or original]

If there is obviously no history - then I may have to fudge it a bit and speak about Pascal's work on his triangle - and never say this was his [but just glide into it] argh... that may be the best way forward

Thanks for your help
Where did you (How did you) come across this triangle? You do have some reference (i.e. it is not completely your creation - yes?!)
 

ink1

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Feb 5, 2018
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Where did you (How did you) come across this triangle? You do have some reference (i.e. it is not completely your creation - yes?!)
No it's not my creation

This is a quote [but the author of a book <Lorayne> suggests the person who wrote this is incorrect - suggesting either he came up with it 'back in the day' or that it was some Pascal variation <Lorayne never elaborates on which or what he means by saying the below quote is incorrect>]


"You explain you are going to form a triangle by adding the two numbers next to each other to get another number. This goes above the two numbers, but if it results in a number over 9, you take 9 away from it.

It is a favourite Harry Lorayne’s yet it was invented years ago by a German. I found it in print in Martin Gardner’s “Mathematical Carnival”, under the chapter dedicated to Pascal’s Triangle "

I have Harry Lorayne's work - but that suggests it's Pascal Triangle [which it isn't] so he's wrong. I don't have Martin Gardner's book - but it looks like from the quote above that he also doesn't give any history to it other than putting it in a chapter about Pascal's Triangle.
 

Jonathan

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Aug 8, 2015
Messages
31
I note the concern of some that we do not give instant answers but as this is recreational maths there is little point going through the "what are your thoughts so far" routine,
I have a copy of Martin Gardner's "Mathematical Carnival".
This is a "magic trick" which Gardner attributes to Harry Lorayne. A spectator is handed a deck of cards from which all 10s and face cards have been removed. The spectator sets out a row of any five cards from the deck. The magician chooses a card and places it face down unseen by others high above the row. Four cards are added to form a row above the five following the rule that each is the sum of the two below it after casting out nines. A row of three is placed above this then a row of two always following the addition rule. The magician then turns over the top card showing that this too is the sum of those below it.
So still left is the puzzle "how does this work".
Hope this is clear, diagram from the book attached.
cardtrick.jpg
 

Dr.Peterson

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Joined
Nov 12, 2017
Messages
2,743
No it's not my creation

This is a quote [but the author of a book <Lorayne> suggests the person who wrote this is incorrect - suggesting either he came up with it 'back in the day' or that it was some Pascal variation <Lorayne never elaborates on which or what he means by saying the below quote is incorrect>]


"You explain you are going to form a triangle by adding the two numbers next to each other to get another number. This goes above the two numbers, but if it results in a number over 9, you take 9 away from it.

It is a favourite Harry Lorayne’s yet it was invented years ago by a German. I found it in print in Martin Gardner’s “Mathematical Carnival”, under the chapter dedicated to Pascal’s Triangle "

I have Harry Lorayne's work - but that suggests it's Pascal Triangle [which it isn't] so he's wrong. I don't have Martin Gardner's book - but it looks like from the quote above that he also doesn't give any history to it other than putting it in a chapter about Pascal's Triangle.
The only place I have access to the Gardner book (though I have others of his) is a library I can't get to quickly. But I found this book that explains the trick, and shows its relation to Pascal's Triangle (which, it appears, Gardner didn't actually explain?). What I learn from this is that the trick itself, which Gardner may have invented or picked up somewhere, is not itself Pascal's triangle, but is derived by applying Pascal's triangle (or, rather, the numbers found in one row of it).

As a result, if you were to mention Pascal's triangle in your introduction to the trick, you would actually be giving away part of its secret, not connecting the trick itself to history.

Jonathan, can you confirm that Gardner doesn't say where he got the trick, and perhaps doesn't even mention how it relates to Pascal's triangle?

Ink1, can you quote what Lorayne actually says relating to the origin of the trick and Pascal? We may be able to see what he really meant if we see his wording.
 

Jonathan

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Dr Peterson, This is Chapter 15 of my Penguin (UK) edition of Mathematical Carnival titled "Pascal's Triangle". Gardner begins the chapter describing this trick mentioning that Harry Lorayne enjoys presenting it. He then describes several properties of Pascal's Triangle linking it to Lorayne's trick at the end where he points out that both share the property on which the trick depends.
Edited to add : Gardner writes that magicians know the trick as "Apex" and that it was originated by a German magician Franz Braun, published 1960 in "Magic" a German magic periodical.
Edited to add : Casting out nines isn't essential to the trick, it makes the use of standard playing cards possible. Using paper and pencil a row of numbers could be written, of any size, the row of any length.
 
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ink1

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Edited to add : Casting out nines isn't essential to the trick, it makes the use of standard playing cards possible. Using paper and pencil a row of numbers could be written, of any size, the row of any length.
Just a small point on this one - for the effect, casting out nines is essential [for the method]

The Lorayne quote is just him saying this:

"WRONG!!"

To the quote in one of my posts above - so it's not clear what [or if all] of that post he thought was wrong [I believe HL is suggesting he and not the German came up with the idea first - which is very possible]

Other than that in his introduction he says this [and only this in regards to origin]

"First I need to show you how to form a 'triangle' (French mathematician Blaise Pascal wrote a treatise on the idea in 1653)"

He doesn't mention it didn't get published until after Pascal died

I still think using an origin and some interesting info - is a great introduction - as is explaining the method they have to employ

I do NOT think it gives away the method [I will be using at all] - even if someone had knowledge of Pascal Triangle. Especially as it's not the actual version they would have been exposed to before.

This is meant to be a slower type presentation. It's also bigger and more seemingly impossible than the card version mentioned above

I believe the card version was a later idea - and the casting out 9's was nothing to do with making it possible to do with playing cards... that's just how it morphed

I'm thinking it was printed in Magic by the German guy [I have no access to that magazine] and then HL updated it with the card version... or maybe his update was the non-card version

Thanks again for all the help

x
 
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